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Theorem elpr 3427
Description: A member of an unordered pair of classes is one or the other of them. Exercise 1 of [TakeutiZaring] p. 15. (Contributed by NM, 13-Sep-1995.)
Hypothesis
Ref Expression
elpr.1  |-  A  e. 
_V
Assertion
Ref Expression
elpr  |-  ( A  e.  { B ,  C }  <->  ( A  =  B  \/  A  =  C ) )

Proof of Theorem elpr
StepHypRef Expression
1 elpr.1 . 2  |-  A  e. 
_V
2 elprg 3426 . 2  |-  ( A  e.  _V  ->  ( A  e.  { B ,  C }  <->  ( A  =  B  \/  A  =  C ) ) )
31, 2ax-mp 7 1  |-  ( A  e.  { B ,  C }  <->  ( A  =  B  \/  A  =  C ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 103    \/ wo 662    = wceq 1285    e. wcel 1434   _Vcvv 2602   {cpr 3407
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-un 2978  df-sn 3412  df-pr 3413
This theorem is referenced by:  prmg  3519  difprsnss  3532  preqr1  3568  preq12b  3570  prel12  3571  pwprss  3605  pwtpss  3606  unipr  3623  intpr  3676  zfpair2  3973  elop  3994  ordtri2or2exmidlem  4277  onsucelsucexmidlem  4280  en2lp  4305  reg3exmidlemwe  4329  xpsspw  4478  acexmidlem2  5540  2oconcl  6086  renfdisj  7239  fzpr  9170  maxabslemval  10232  isprm2  10643  bj-zfpair2  10859
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